EPFO EO/AO 2023 General Ability Test · Question 72 of 120

Correct — A, (a) 116. The question is two definitions dressed as a puzzle. 'The smallest natural number that is divisible by both 24 and 30' is the definition of the least common multiple, and 'the largest natural number that divides both 36 and 100' is the definition of the highest common factor. So x = LCM(24, 30) and y = HCF(36, 100). Take the prime factorisations. 24 = 2³ × 3 and 30 = 2 × 3 × 5. The LCM takes the highest power of each prime that appears in either: 2³ × 3 × 5 = 120. So x = 120. As a check, HCF(24, 30) = 2 × 3 = 6, and LCM × HCF = 120 × 6 = 720 = 24 × 30, which is the identity every such pair satisfies. Now 36 = 2² × 3² and 100 = 2² × 5². The HCF takes the lowest power of each prime that appears in both. The only prime common to the two is 2, and it appears squared in each, so HCF = 2² = 4. There is no 3 in 100 and no 5 in 36, so nothing else survives. So y = 4. Hence x − y = 120 − 4 = 116, which is option (a). There is also a check that kills the other three options without computing anything at all. Whatever y turns out to be, it is a natural number and therefore at least 1, so x − y is strictly less than x = 120. Options (b), (c) and (d) are 124, 128 and 132 — every one of them larger than 120. Once the LCM is known to be 120, only one option is even arithmetically possible. That is worth internalising as a habit: in a question of the form 'find A − B' where A is quickly computed and B is positive, any option at or above A can be struck out on sight.

• (b)124 — This is 120 + 4 — the right LCM and the right HCF, combined with the wrong sign. The stem asks for x − y, and a candidate reading quickly under time pressure adds instead. It is the most common wrong answer to this shape of question, and it is also impossible on inspection: x − y cannot exceed x, and 124 is larger than 120.
• (c)128 — This is 120 + 8, and 8 is the sort of number a hurried candidate produces for the HCF of 36 and 100 by looking only at 100 — 8 divides 100? It does not, and that is precisely the check that fails: 100 ÷ 8 = 12.5, and in any case 8 does not divide 36. A common factor has to divide both numbers, and the largest that divides 36 and 100 alike is 4. Like option (b), this figure also exceeds the LCM and so cannot be a difference of the required form.
• (d)132 — This is 120 + 12. Twelve is a real factor of 36 and a tempting one, but it does not divide 100 — 100 ÷ 12 is not a whole number — so it cannot be a common factor of the two at all. Options (c) and (d) are a matched pair of traps: one offers a number that divides 100 but not 36, the other a number that divides 36 but not 100. The HCF must satisfy both, which is why the answer collapses to 2² = 4.

Every question of this family is settled by prime factorisation. For the highest common factor, take each prime that appears in all the numbers and raise it to the lowest power it has in any of them. For the least common multiple, take each prime that appears in any of them and raise it to the highest power it has in any of them. Two consequences are worth carrying. First, for exactly two numbers, LCM × HCF = the product of the numbers — so having found one, the other follows by division; this fails for three or more numbers, which is a favourite trap. Second, the HCF of a set never exceeds the smallest member and the LCM is never smaller than the largest, which gives a free sanity check on every answer. The word problems this appears in are usually one of three: the smallest quantity that can be divided exactly (LCM), the largest measure that fits exactly into several quantities (HCF), or bells or lights that coincide after an interval (LCM again). The present question is unusual only in that it states the two definitions outright instead of hiding them in a story.

This is a one-line computation dressed up in words, and its purpose in the paper is to test whether the candidate can translate a definition into an operation quickly. EPFO's quantitative block contains several such items and they are the ones to bank early in the paper, before the reasoning and data questions eat the clock. The habit rewarded is recognising the definition — 'smallest number divisible by both' and 'largest number that divides both' — rather than trying to search for the number.

• The least common multiple of 24 and 30 is 120: 24 = 2³ × 3, 30 = 2 × 3 × 5, and the LCM takes the highest power of each prime, 2³ × 3 × 5.
• The highest common factor of 36 and 100 is 4: 36 = 2² × 3², 100 = 2² × 5², and the only shared prime is 2, squared in both.
• For any two numbers, LCM × HCF equals the product of the numbers — here 120 × 6 = 720 = 24 × 30 for the pair 24 and 30.
• That identity does not extend to three or more numbers, so it cannot be used to recover an HCF from an LCM in a three-number problem.
• The HCF of a set of numbers never exceeds the smallest of them; the LCM is never less than the largest of them.
• x − y = 120 − 4 = 116; and since y is a natural number, x − y must be strictly less than 120, which rules out every other option here.

• Adding instead of subtracting. The stem asks for x − y, and 124 is on the option list to catch exactly that.
• Offering a factor of only one of the two numbers as the HCF. 8 divides 100 but not 36; 12 divides 36 but not 100.
• Using LCM × HCF = product for more than two numbers. The identity holds only for a pair.
• Confusing which is which. Smallest number divisible by both is the LCM; largest number dividing both is the HCF.

EO/AO asks HCF and LCM either as a bare computation like this one or wrapped in a measurement or timing story. Both are answered from the prime factorisations in a few seconds, and both reward the two sanity checks — the HCF cannot exceed the smallest number given, and the LCM cannot be smaller than the largest.

No directly related past PYQ was found.

• practice — not a real PYQ

  If the HCF of two numbers is 12 and their LCM is 180, and one of the numbers is 36, then the other number is :

  • (a)45
  • (b)60
  • (c)72
  • (d)90

  Answer(b) 60

• practice — not a real PYQ

  What is the largest natural number that divides both 48 and 180 ?

  • (a)6
  • (b)12
  • (c)18
  • (d)24

  Answer(b) 12